In a ` DeltaA B C ,\ ` prove that `sin^3Acos(B-C)+sin^3B cos(C-A)+sin^3\ C cos(A-B)=3\ sin A\ sin B\ sin C`

Explanation

LHS =`sin ^{3} A cos (B-C)+sin ^{3} B(C-A)+sin ^{3} cos (A-B)`

`=sin ^{2} A-sin (B+C) cos (B-C)+sin ^{2} B sin (C+A) cos (C-A)+sin ^{2} C sin (A+B) cos`

`(A-B)[therefore A+B+C=pi]`

`=frac{1}{2}[sin ^{2} A(sin 2 B+sin 2 C)+sin ^{2} B(sin 2 C+sin 2 A)+sin ^{2}`.

`C(sin 2 A+sin 2 B)][therefore sin C+sin D=2 sin frac{C+D}{2} cos frac{C+D}{2}]`

`=frac{2}{2}[sin ^{2} A sin B cos B+sin ^{2} A sin C cos C+sin ^{2} B sin C cos C+sin ^{2} B sin A cos A+sin ^{2}.` `.(sin A cos A+sin ^{2} C sin B cos B)]`

`=sin A sin B(sin A cos B+cos A sin B)+sin B sin C(sin B cos C+cos B sin C)+sin C sin
A(sin C cos A+cos C sin A)`

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